The difference is entirely explained by worse insulin sensitivity (a result of being very fat, esp. in the abdominal area). The task is 2-Back (which is easy). The specific brain region is definitely expected to activate in short-term memory tasks, and is associated with performance on them.

There are plenty of reasons to believe insulin resistance (usually due to obesity) damages the brain. Resistance means more insulin is produced, which harms the blood vessels in the brain, which exacerbates dementia (for the same reason, heavy smoking is bad).

insulin is an important factor for the successful operation of several processes related to learning and memory such as glucose metabolism, neurotransmitter release, and long-term potentiation (10,29). Insulin from the periphery has been shown to cross the blood–brain barrier in a dose-dependent manner (32). However, chronic hyperinsulinemia and insulin resistance are believed to cause insulin receptors on the blood–brain barrier to downregulate, inducing an insulin deficit state in the brain and negatively affecting neurophysiological processes critical to cognitive functioning (29). In keeping with these findings, rats with diet-induced obesity and insulin resistance display impairments in spatial memory and reduced hippocampal synaptic plasticity (30). Similarly, humans with diabetes mellitus exhibit impaired declarative memory and hippocampal atrophy (33). Last but not least, insulin resistance is implicated in the pathogenesis of Alzheimer’s type dementia through stimulation of cellular release of amyloid-β and reduction in amyloid-β clearance (29).